In this paper we analyze the vibrations of a bridge modeled as an isotropic plate with all sides simply supported under the effect of a moving load due to a full vehicle having seven degrees of freedom. A mathematical model of the bridge is obtained by applying Lagrange's formulation to orthogonal mode shapes and the non-conservative moving forces. The modern materials used in the construction of the bridges satisfy isotropic conditions mostly, and they can be modeled as rectangular plates with four sides simply supported. The time responses of the mid-span and quarter-span of the bridge are obtained. The transverse vibration of the bridge and the body bounce, pitch, and roll of the vehicle are presented for different vehicle speeds. Finally, the bending moment at the mid-span is presented for different vehicle speeds to aid in the structural design of the bridge.